edited by
8,794 views
37 votes
37 votes

Let $f$ be a function from a set $A$ to a set $B$, $g$ a function from $B$ to $C$, and $h$ a function from $A$ to $C$, such that $h(a) = g(f(a))$ for all $a ∈ A.$ Which of the following statements is always true for all such functions $f$ and $g$?

  1. $g$ is onto $\implies$ $h$ is onto
  2. $h$ is onto $\implies$ $f$ is onto
  3. $h$ is onto $\implies$ $g$ is onto
  4. $h$ is onto $\implies$ $f$ and $g$ are onto
edited by

8 Answers

2 votes
2 votes

If condition p --> q is true then its contrapositive ( ~q --> ~p ) is also true

above example is if g is not on to then 'h' is also not onto

so its contrapositive is also true that is { h is onto --> g is onto}

0 votes
0 votes

…....………………………………………………

0 votes
0 votes

The correct option is C h is onto ⇒ g is onto
Given, h=g(f(x))=g.f
Consider the following arrow diagram

From above diagram it is clear that
g is not onto ⇒h=g.f is also not onto, since the co-domain of g is same as the co-domain of g.f.
The contrapositive version of the above implication is
h is onto ⇒g is onto
which also has to be true since direct ≡ contrapositive.
So option (c) is true.

–5 votes
–5 votes
option D
Answer:

Related questions

31 votes
31 votes
7 answers
1
44 votes
44 votes
10 answers
2